Generating family invariants for Legendrian links of unknots

Abstract

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in ${ℝ}^3$. It is shown that the unknot with maximal Thurston–Bennequin invariant of $-1$ has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for $2$–component Legendrian links where each component is a maximal unknot. Techniques are developed to compute these polynomials, and computations are done for two families of Legendrian links: rational links and twist links. The polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links.